Pierre Arnoux: Algebraic numbers and automorphisms of free groups

This is a joint work with Maki Furukado, Edmund Harriss and Shunji Ito.

We consider a simple example of automorphism of the free group on 4 generators, whose associated matrix has 4 distinct complex eigenvalues, two of them of modulus larger than 1, and the other 2 of modulus smaller than 1 (non-Pisot case). Using this generator, we show how to build substitutions polygonal tilings of the contracting plane and the expanding plane of the matrix. We prove that these substitution tilings can be lifted in a unique way to stepped surfaces approximating each of these planes. The vertices of these stepped surfaces can be projected to atomic surfaces, compact sets with fractal boundary contained in the other plane. We prove that these tilings can be refined to exact self-similar tilings, whose tiles have fractal boundaries, and can be obtained by iteration, or from the atomic surface. Using the self-similar tiling, on can build a numeration system; the natural extension of the map associated with this number system is the linear map obtained by abelianization of the free group automorphism. This gives an explicit Markov partition of this toral automorphism. The fractal domains can be used to define a pseudo-group of translations which gives transeversal dynamics in the sense of Vershik or numeration systems in the sense of Kamae.


G. Rauzy.
Nombres algébriques et substitutions.
Bull. Soc. Math. France, 110(2):147-178, 1982.

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